int main()
{
slong iter;
flint_rand_t state;
flint_printf("atan_arf_bb....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 5000; iter++)
{
arb_t x, y, z;
slong prec, prec2;
arb_init(x);
arb_init(y);
arb_init(z);
prec = 2 + n_randint(state, 8000);
arb_randtest(x, state, 1 + n_randint(state, 8000), 3);
mag_zero(arb_radref(x));
if (n_randint(state, 2))
arb_mul_2exp_si(x, x, 1 + n_randint(state, 40));
else
arb_mul_2exp_si(x, x, -n_randint(state, 1.5 * prec));
if (!arf_is_special(arb_midref(x)))
prec2 = prec + 100 + 2 * (-ARF_EXP(arb_midref(x)));
else
prec2 = prec + 100;
arb_atan_arf_via_mpfr(y, arb_midref(x), prec2);
arb_atan_arf_bb(z, arb_midref(x), prec);
if (!arb_contains(z, y))
{
flint_printf("FAIL: containment\n\n");
flint_printf("prec = %wd\n\n", prec);
flint_printf("x = "); arb_printd(x, 50); flint_printf("\n\n");
flint_printf("y = "); arb_printd(y, 50); flint_printf("\n\n");
flint_printf("z = "); arb_printd(z, 50); flint_printf("\n\n");
abort();
}
if (arb_rel_accuracy_bits(z) < prec - 2)
{
flint_printf("FAIL: poor accuracy\n\n");
flint_printf("prec = %wd, acc = %wd\n\n", prec, arb_rel_accuracy_bits(z));
flint_printf("x = "); arb_printd(x, 50); flint_printf("\n\n");
flint_printf("y = "); arb_printd(y, 50); flint_printf("\n\n");
flint_printf("z = "); arb_printd(z, 50); flint_printf("\n\n");
abort();
}
arb_atan_arf_bb(x, arb_midref(x), prec);
if (!arb_overlaps(x, z))
{
flint_printf("FAIL: aliasing\n\n");
abort();
}
arb_clear(x);
arb_clear(y);
arb_clear(z);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
arb_atan_arf(arb_t z, const arf_t x, slong prec)
{
if (arf_is_special(x))
{
if (arf_is_zero(x))
{
arb_zero(z);
}
else if (arf_is_pos_inf(x))
{
arb_const_pi(z, prec);
arb_mul_2exp_si(z, z, -1);
}
else if (arf_is_neg_inf(x))
{
arb_const_pi(z, prec);
arb_mul_2exp_si(z, z, -1);
arb_neg(z, z);
}
else
{
arb_indeterminate(z);
}
}
else if (COEFF_IS_MPZ(*ARF_EXPREF(x)))
{
if (fmpz_sgn(ARF_EXPREF(x)) < 0)
arb_atan_eps(z, x, prec);
else
arb_atan_inf_eps(z, x, prec);
}
else
{
slong exp, wp, wn, N, r;
mp_srcptr xp;
mp_size_t xn, tn;
mp_ptr tmp, w, t, u;
mp_limb_t p1, q1bits, p2, q2bits, error, error2;
int negative, inexact, reciprocal;
TMP_INIT;
exp = ARF_EXP(x);
negative = ARF_SGNBIT(x);
if (exp < -(prec/2) - 2 || exp > prec + 2)
{
if (exp < 0)
arb_atan_eps(z, x, prec);
else
arb_atan_inf_eps(z, x, prec);
return;
}
ARF_GET_MPN_READONLY(xp, xn, x);
/* Special case: +/- 1 (we require |x| != 1 later on) */
if (exp == 1 && xn == 1 && xp[xn-1] == LIMB_TOP)
{
arb_const_pi(z, prec);
arb_mul_2exp_si(z, z, -2);
if (negative)
arb_neg(z, z);
return;
}
/* Absolute working precision (NOT rounded to a limb multiple) */
wp = prec - FLINT_MIN(0, exp) + 4;
/* Too high precision to use table */
if (wp > ARB_ATAN_TAB2_PREC)
{
arb_atan_arf_bb(z, x, prec);
return;
}
/* Working precision in limbs */
wn = (wp + FLINT_BITS - 1) / FLINT_BITS;
TMP_START;
tmp = TMP_ALLOC_LIMBS(4 * wn + 3);
w = tmp; /* requires wn+1 limbs */
t = w + wn + 1; /* requires wn+1 limbs */
u = t + wn + 1; /* requires 2wn+1 limbs */
/* ----------------------------------------------------------------- */
/* Convert x or 1/x to a fixed-point number |w| < 1 */
/* ----------------------------------------------------------------- */
if (exp <= 0) /* |x| < 1 */
{
reciprocal = 0;
/* todo: just zero top */
flint_mpn_zero(w, wn);
/* w = x as a fixed-point number */
error = _arf_get_integer_mpn(w, xp, xn, exp + wn * FLINT_BITS);
}
else /* |x| > 1 */
{
slong one_exp, one_limbs, one_bits;
mp_ptr one;
reciprocal = 1;
one_exp = xn * FLINT_BITS + wn * FLINT_BITS - exp;
flint_mpn_zero(w, wn);
/* 1/x becomes zero */
if (one_exp >= FLINT_BITS - 1)
{
/* w = 1/x */
one_limbs = one_exp / FLINT_BITS;
one_bits = one_exp % FLINT_BITS;
if (one_limbs + 1 >= xn)
{
one = TMP_ALLOC_LIMBS(one_limbs + 1);
flint_mpn_zero(one, one_limbs);
one[one_limbs] = UWORD(1) << one_bits;
/* todo: only zero necessary part */
flint_mpn_zero(w, wn);
mpn_tdiv_q(w, one, one_limbs + 1, xp, xn);
/* Now w must be < 1 since x > 1 and we rounded down; thus
w[wn] must be zero */
}
}
/* todo: moderate powers of two would be exact... */
error = 1;
}
/* ----------------------------------------------------------------- */
/* Table-based argument reduction */
/* ----------------------------------------------------------------- */
/* choose p such that p/q <= x < (p+1)/q */
if (wp <= ARB_ATAN_TAB1_PREC)
q1bits = ARB_ATAN_TAB1_BITS;
else
q1bits = ARB_ATAN_TAB21_BITS;
p1 = w[wn-1] >> (FLINT_BITS - q1bits);
/* atan(w) = atan(p/q) + atan(w2) */
/* where w2 = (q*w-p)/(q+p*w) */
if (p1 != 0)
{
t[wn] = (UWORD(1) << q1bits) + mpn_mul_1(t, w, wn, p1);
flint_mpn_zero(u, wn);
u[2 * wn] = mpn_lshift(u + wn, w, wn, q1bits) - p1;
mpn_tdiv_q(w, u, 2 * wn + 1, t, wn + 1);
error++; /* w2 is computed with 1 ulp error */
}
/* Do a second round of argument reduction */
if (wp <= ARB_ATAN_TAB1_PREC)
{
p2 = 0;
}
else
{
q2bits = ARB_ATAN_TAB21_BITS + ARB_ATAN_TAB22_BITS;
p2 = w[wn-1] >> (FLINT_BITS - q2bits);
if (p2 != 0)
{
t[wn] = (UWORD(1) << q2bits) + mpn_mul_1(t, w, wn, p2);
flint_mpn_zero(u, wn);
u[2 * wn] = mpn_lshift(u + wn, w, wn, q2bits) - p2;
mpn_tdiv_q(w, u, 2 * wn + 1, t, wn + 1);
error++;
}
}
/* |w| <= 2^-r */
r = _arb_mpn_leading_zeros(w, wn);
/* N >= (wp-r)/(2r) */
N = (wp - r + (2*r-1)) / (2*r);
/* Evaluate Taylor series */
_arb_atan_taylor_rs(t, &error2, w, wn, N, 1);
/* Taylor series evaluation error */
error += error2;
/* Size of output number */
tn = wn;
/* First table lookup */
if (p1 != 0)
{
if (wp <= ARB_ATAN_TAB1_PREC)
mpn_add_n(t, t, arb_atan_tab1[p1] + ARB_ATAN_TAB1_LIMBS - tn, tn);
else
mpn_add_n(t, t, arb_atan_tab21[p1] + ARB_ATAN_TAB2_LIMBS - tn, tn);
error++;
}
/* Second table lookup */
if (p2 != 0)
{
mpn_add_n(t, t, arb_atan_tab22[p2] + ARB_ATAN_TAB2_LIMBS - tn, tn);
error++;
}
/* pi/2 - atan(1/x) */
if (reciprocal)
{
t[tn] = LIMB_ONE - mpn_sub_n(t,
arb_atan_pi2_minus_one + ARB_ATAN_TAB2_LIMBS - tn, t, tn);
/* result can be >= 1 */
tn += (t[tn] != 0);
/* error of pi/2 */
error++;
}
/* The accumulated arithmetic error */
mag_set_ui_2exp_si(arb_radref(z), error, -wn * FLINT_BITS);
/* Truncation error from the Taylor series */
mag_add_ui_2exp_si(arb_radref(z), arb_radref(z), 1, -r*(2*N+1));
/* Set the midpoint */
inexact = _arf_set_mpn_fixed(arb_midref(z), t, tn, wn, negative, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
TMP_END;
}
}
